Aguero scored twice, with Fabian Delph and David Silva also on target, as City beat Crystal Palace 4-0, and Terry atoned for an earlier own goal to earn Chelsea an improbable point at Stamford Bridge.
City provisionally top the standings on goal difference above previous leaders Arsenal, who visit Stoke City on Sunday, and Leicester City, who travel to Aston Villa later on Saturday.
City manager Manuel Pellegrini made a bold approach at a snowy Etihad Stadium by aligning Aguero up front with Kelechi Iheanacho in a 4-4-2 formation.
After City goalkeeper Joe Hart had parried an early header by Damien Delaney, Delph opened the scoring in the 22nd minute with a skidding 30-yard shot that beat Palace goalkeeper Wayne Hennessey at his left-hand post.
Aguero made it 2-0 shortly before half-time with an effort from outside the box that was deflected past Hennessey by the head of Scott Dann.
The hosts had left-back Aleksandar Kolarov stretchered off in the 52nd minute with an apparent left calf problem, but Aguero struck again 22 minutes from time by tapping in a low cross from Kevin De Bruyne.
The Argentine could have had a hat-trick in the 84th minute, but after sprinting clear of the Palace defence he unselfishly squared for Silva to tap in City’s fourth goal.
Champions Chelsea remain six points above the bottom three after interim manager Guus Hiddink avoided the first defeat of his second stint at the club by the narrowest of margins.
Everton went 2-0 up early in the second half as Terry scored an inelegant 50th-minute own goal and Kevin Mirallas followed up with a slick turn and shot, both goals created by Leighton Baines.
– Spurs back in groove –
Cesc Fabregas brought Chelsea back into the game, freeing Diego Costa to round Tim Howard and reduce the arrears in the 64th minute before equalising with a deflected shot less than two minutes later.
After Costa had limped off injured, centre-back Ramiro Funes Mori put Everton ahead in the 91st minute from Gerard Deulofeu’s cross, only for Terry to flick home a last-gasp equaliser despite an apparent offside.
Tottenham Hotspur tasted victory in 2016 at the fourth attempt with a one-sided 4-1 home success over Sunderland in which Christian Eriksen scored twice.
Sunderland, chasing a third straight league win, took a 40th-minute lead when Adam Johnson released Patrick van Aanholt to score.
But Eriksen equalised 91 seconds later, following up to score after Sunderland league debutant Jordan Pickford had thwarted Harry Kane, and goals from Mousa Dembele, Eriksen and Kane, with a penalty, gave Spurs victory.
Spurs, who occupy the fourth Champions League berth, opened up a five-point lead over Manchester United, who visit Liverpool on Sunday, and a four-point lead over West Ham United, who lost 2-1 at Newcastle United.
Jonjo Shelvey shone on his Newcastle debut following his £12 million (16 million euros, $17.3 million) move from Swansea City, sparking the moves that led to early goals for Ayoze Perez and Georginio Wijnaldum.
Substitute Nikica Jelavic hit back for West Ham in the 49th minute after intercepting a back-pass from Chancel Mbemba, but Newcastle held on to escape the relegation zone at the expense of Shelvey’s old club Swansea.
Bournemouth pulled five points clear of danger by beating fellow strugglers Norwich City 3-0, with Dan Gosling, Charlie Daniels and £10 million record signing Benik Afobe on target.
Meanwhile, James Ward-Prowse scored a fine free-kick and a penalty as Southampton, watched by new striker Charlie Austin, won 3-0 at home to West Bromwich Albion, with Serbian winger Dusan Tadic also scoring.
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